While the study of optimal transport on metric graphs is relatively new [1]}, the study of nodal sets of special functions on metric graphs is a well established field, see e.g., [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, as well as [9]} and the references therein. In its heart, the question is a natural extension of Sturm's oscillations Theory: given the \(N\) -th Laplacian eigenfunction on a certain metric graph, how many zeroes does it have?
